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A recent method called asymptotic Taylor expansion (ATEM) is applied to determine the analytical expression for eigenfunctions and numerical results for eigenvalues of the Schrödinger equation for the bistable potentials. Optimal truncation of the Taylor series gives a best possible analytical expression for eigenfunctions and numerical results for eigenvalues. It is shown that the results are obtained by a simple algorithm constructed for a computer system using symbolic or numerical calculation. It is observed that ATEM produces excellent results consistent with the existing literature.

Energy eigenvalues of quartic and sextic type anharmonic potentials are obtained by using an alternative method called asymptotic Taylor expansion method (ATEM) which is an approximate approach based on the asymptotic Taylor series expansion of a function. It is shown that the energy eigenvalues found by ATEM are in excellent agreement with the existing results.

The effect for different types of scattering on the critical half thickness in slab geometry for one speed neutron transport equation is studied for isotropic, linearly anisotropic and quadratic anisotropic scatterings. An extensive numerical survey is carried out for the critical thickness in order to provide the effect of the different scattering types. The numerical results are obtained by PN, TN and UN methods. The PN method is the Legendre polynomial solution that is accepted as the exact result for the neutron transport theory calculations and the UN and TN methods are the types of Chebyshev polynomials. Critical thickness values are calculated by using Mark boundary condition. Results are compared with the literature

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